Why are Beowulf and Antigone both considered heroes? What makes them heroic?

Beowulf and Antigone are indeed both heroic characters. Beowulf is heroic in the tradition of myth and legend. As a young warrior, he is strong, courageous, and fierce in battle; he is also a doer of noble deeds:

[Beowulf was] the strongest of the Geats--greater

And stronger than anyone anywhere in this world--

[Beowulf] heard how Grendel filled nights with horror

And quickly commanded a boat fitted out,

Proclaiming that he'd go to the famous king.

Would sail across the sea to Hrothgar,

Now when help was needed.

Once he faces Grendel, who can smash the bodies of thirty men at one time and carry them away, Beowulf tears off the monster's arm, then follows him to the bottom of the sea to kill him. The epic poem is filled with acts of heroism such as these and concludes as an aged King Beowulf is mortally wounded when he battles a dragon.

Antigone's heroism is of a quieter nature, but just as impressive. She remains true to her moral principles by defying King Creon's order against burying the body of her brother. Antigone buries her brother's body, even though she knows the punishment for her crime is death. When confronted with her crime, she stands up to Creon's wrath, refusing to apologize or beg for her own life:

Creon, what more do you want than my death . . . . Then I beg you: kill me.

This talking is a great weariness: your words

Are distasteful to me, and I am sure that mine

Seem so to you. And yet they should not seem so:

I should have praise and honor for what I have done.

Antigone does not fear death; when she is entombed by Creon, buried alive, she takes her own life by hanging herself. The heroism in Antigone is that she chooses to die rather than live without honor and integrity:

This death of mine

Is of no importance; but if I had left my brother

Lying in death unburied, I should have suffered.

Now I do not.

Beowulf and Antigone both live and die with courage.